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We study the area distribution of closed walks of length $n$, beginning andending at the origin. The concept of area of a walk in the square lattice isgeneralized and the usefulness of the new concept is demonstrated through asimple argument. It is concluded that the number of walks of length $n$ andarea $s$ equals to the coefficient of $z^s$ in the expression$(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a specialgroup ring $R[x,y,z]$. A polynomial time algorithm for calculating thesevalues, is then concluded. Finally, the provided algorithm and the results ofimplementation are compared with previous works.

Enumeration of closed random walks in the square lattice according to their areas